Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

Hello guys any help would be appreciated for this question.

Show that the general solution of y=x*p + f(p) is y=cx+f(c), where p=dy/dx and c is an arbitrary constant. Also show that there is a 2nd solution obeying the diff. equation d/dp(f(p)) + x =0. Finally find the singular solution obeying both of the diff equations.

Obviously the first equation is not a separable equation. Also can't use the integrating factor since we have a function of the derivative in our equation. Can anyone tell me how do we solve this type of differential equations since is the first time I see this type? By direct differentiation of the given solution is easy to show that it is a solution but I don't think so this is a way. And after solving the first equation how do I proceed for the other parts?

Thanks in advance for any help!

re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

That is actually the way to solve this ODE. If you differentiate wrt $\displaystyle x$ you get

$\displaystyle p = p + xp' + f'(p)p'$

or

$\displaystyle p'\left( f'(p) + x\right) = 0$ - two cases

$\displaystyle (1)$ $\displaystyle p ' = 0 $ gives $\displaystyle y = ax + b$ (subs this back into your original ODE to find conditions on $\displaystyle a$ and $\displaystyle b$)

$\displaystyle (2)$ $\displaystyle f'(p) +x = 0$ is the second choice

Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

Thank you very much! Substituting in the original ODE I get b=f(a), so from the first moment should I have put that y=cx+b and then find that b=f(c)? Also how do I find a singular solution obeying the two equations (y=x*p + f(p), d/dp (f(p)) + x= 0)

Thanks again very much!

Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

I am still stucked with this exercise, can anyone please help me about how do I find a singular solution obeying the two equations (y=x*p + f(p), d/dp (f(p)) + x= 0)? Thank you very much!

Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

Suppose that $\displaystyle f(p)$ is some know function then

$\displaystyle x = -f'(p)$

and from the first

$\displaystyle y = p x + f(p) = f(p) - p f'(p).$

So now you have $\displaystyle x $ and $\displaystyle y$ in terms of the parameter $\displaystyle p$ (i.e. you have a parametric solution).

Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

thank you very much, your help is appreciated!